It looks to me like your calculus and physics textbooks are talking about slightly different things. Your calculus book is giving you a discussion of formal vector fields, whereas from the notation, I take it that your physics text is just considering some simple scalar work and energy ideas.

The matter of the negative sign in an equation like your W = -ΔU is a choice that represents who is doing work on who. try to find in your physics book the exactly what it means by 'W'; is that the work that the system with potential energy U is doing, or is it the work being performed on that system? Judging by the placement of the negative sign you were wondering about, it looks like that equation is saying "the work W done BY the system is equal to the negative of the change in its potential energy'; qualitatively this makes some sense, since the system has to expend some energy to do some work on something external to it.

Let's take an example. Suppose you drop from rest a ball of mass m and let it fall for 3 seconds. You want to find the amount of work that gravity does on the object. There are three ways you could proceed.

1) Use W=[itex]\int[/itex][itex]\vec{F}[/itex][itex]\cdot[/itex]d[itex]\vec{r}[/itex] the line integral method. (Note that this method can be made simpler by first finding the potential function f such that gradF = f and then calculating the change in f. In other words, use W=Δf.)
2) Use W=ΔK the work-energy theorem.
3) Use W=-ΔU where U=mgy

I'll spare you the details, but either way the work comes out to 4.5mg2. Note that in the first method I used f=-mgy. So in other words U=-f. Why is U defined this way? It seems to me that things would be a whole lot less confusing if U was just defined to be equal to f.